Comparison of two volume integral equation formulations for solving electromagnetic scattering by inhomogeneous dielectric objects

Two types of volume integral equations are applied to solve electromagnetic scattering by inhomogeneous dielectric objects. The first one is the classical volume integral equation involving the electric flux density as unknown (D-formulation). This equation is discretized using Galerkin's method and divergence conforming basis function. The volume integral equation can also be written for the electric field (E-formulation). The drawback of the conventional E-formulation, however, is that when discretized with Galerkin's method and curl conforming basis functions additional surface integrals appear on the boundaries of the elements if the gradient is moved to the testing function. In order to avoid this, in this paper a novel E-formulation is introduced. In the following, we first present two formulations of the volume integral equations that, after discretized with Galerkin's method, contain only weakly singular kernels, and then present numerical results and compare the solution accuracy and the number of iterations of the bi-conjugate gradient method.